Vestigial nematicity from spin and/or charge order in the cuprates

Nematic order has manifested itself in a variety of materials in the cuprate family. We propose an effective field theory of a layered system with incommensurate, intertwined spin- and charge-density wave (SDW and CDW) orders, each of which consists of two components related by $C_4$ rotations. Using a variational method (which is exact in a large-$N$ limit), we study the development of nematicity from partially melting those density waves by either increasing temperature or adding quenched disorder. As temperature decreases we first find a transition to a single nematic phase, but depending on the range of parameters (e.g., doping concentration) the strongest fluctuations associated with this phase reflect either proximate SDW or CDW order. We also discuss the changes in parameters that can account for the differences in the SDW-CDW interplay between the 214 family and the other hole-doped cuprates.

Figure 1

(a) Mean-field (ignoring spatial fluctuations) phase diagram for the parameters defined in Sec. 2a and Eqs. (2.4)$\sim$(2.7) taken to be ${\alpha }_{s0}={\alpha }_{\rho 0}=1.1,T_{\text{sdw}0}=1.3,T_{\text{cdw}0}=0.7$. Solid lines mark the second-order mean-field phase transitions. Red (blue) dot-dashed line marks the SDW (CDW) transition when there are no interaction terms between SDW and CDW, i.e., $v=v^{\prime }=0$. (b) Large-$N$ phase diagram with the same parameters as in (a), but including spatial fluctuations, and obtained from the variational approach introduced in Sec. 3. The solid lines denote second order phase transitions and the four phase boundaries meet at a tetracritical point. If we had taken parameters such that $v>v^{\prime }>\sqrt{u_s{u}_{\rho }}$, the narrow range of SDW and CDW coexistence at low temperature would have been replaced by a first order transition terminating at a bicritical point. The transition line to the (extremely narrow) vestigial nematic phase has been shifted upward by $\mathrm{\Delta }T=0.05$ for graphical clarity.

Figure 2

Large-$N$ phase diagram with input parameters same as in Fig. 1 but with “weak” disorder, $\sigma =0.1$. All transitions are second order. The nematic transition line is again shifted upward by $\mathrm{\Delta }T=0.05$ for graphical clarity. Consistent with general theorems, the CDW phase has been eliminated, but the nematic and SDW transition temperatures have only been decreased to almost unnoticeable degree relative to the zero disorder case in Fig. 1.

Figure 3

Large-$N$ phase diagram with the same parameters as in Fig. 1 but with “moderate” disorder, $\sigma =0.5$. Dashed and solid lines denote first- and second-order transitions, respectively, and the black dots are tricritical points (see text). The nematic transition line is again shifted upward by $\mathrm{\Delta }T=0.05$ for graphical clarity. Notice the observable suppression of the nematic transition temperature compared to Figs. 1 and 2.